The total number of positive integral solutions (x,y,z) such that xyz=24 is:
Solutions · 1
Official solution
x.y.z=24
\[x.y.z = {2^3}.\,{3^1}\]
Three 2 has to be distributed among x, y and z
Each may receive none, one or two
\[\therefore\] Number of ways = 3+3−1C3−1 = 5C2 ways
Similarly one 3 has to be distributed among x, y and z
\[ \therefore \] Number of ways = 1+3−1C3−1 = 3C2 ways
Total ways = \[^5{C_2}\,.{\,^3}{C_2}\] = 30